Pen & Paper

Due: Sunday, Nov 22nd 2020, 23:59

No late submissions / grace days allowed!!

The following questions are a compilation of exercises from the books this class is based upon.

Task 1

As geometric data passes through the viewing pipeline, a sequence of rotations, translations, scaling, and a projection transformation are applied to the vectors that determine the cosine terms in the Phong reflection model.
  a) Which, if any, of these operations preserve(s) the angles between the vectors? (1 point)
  b) Estimate the amount of extra calculations required for Phong shading as compared to Gouraud shading, taking into account the previously mentioned transformations. (1 point)

Task 2

Find the projection of a point onto the plane ax + by + cz + d = 0 from a light source located at infinity in the direction (dx,dy,dz). (2 points)

Task 3

Consider a highly reflective sphere centered at the origin with a unit radius. If a viewer is located at P, describe the points she would see reflected in the sphere at a point on its surface. (2 points)

Task 4

For many Virtual Reality installations, the COP can be at any point and the projection plane can be at any orientation. Derive the projection matrix for this general case. (2 points)

Task 5

Find four points equidistant from one another on a unit sphere. These points determine a tetrahedron. Find the general solution with explanation how you developed it. (Hint: You can arbitrarily let one of the points be at (0, 1, 0) and let the other three be in the plane y = −d, for some positive value of d). (2 points)